Let $X$ be a compact Riemann surface and $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ be a ${\Bbb R}$-divisor on $X$ such that $\theta_j\geq 0$ and $P_1,\cdots,P_n$ are $n$ distinct points on $X$. We call $ds^2$ a conformal metric representing $D$ if $ds^2$ is a smooth conformal metric on $X\backslash {\rm Supp}\, D:=X\backslash \{P_1,\cdots, P_n\}$ and in a neighborhood $U_j$ of $P_j$, $ds^2$ has form $e^{2u_j}\,|dz|^2$, where $z$ is a local complex coordinate defined in $U_j$ centered at $P_j$, as $\theta_j>0$ the real valued function $u_j-(\theta_j-1)\,\ln\,|z|$ is continuous in $U_j$, and as $\theta_j=0$ the real valued function $u_j+\ln\,|z|+\ln\,\big(-\ln\,|z|\big)$ is continuous in $U_j$. We also call that $ds^2$ has cone singularity of angle $2\pi\theta_j$ at $P_j$ as $\theta_j>0$, and has cusp singularity at $P_j$ as $\theta_j=0$. Note that $ds^2$ has finite area near a cone or cusp singularity. It is well known that if a conformal flat or spherical (positive constant curvature) metric has finite area, then its isolated singularities must be cone singularities. The Uniformization Theory gives a class of conformal hyperbolic metrics on Riemann surfaces with mixed cusp singularities and cone singularities of angles in $\{2\pi/2,\,2\pi/3,\, 2\pi/4,\cdots\}$. By the Gauss-Bonnet formula, if $ds^2$ is a conformal hyperbolic metric representing $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$, then by Gauss-Bonnet formula there holds $\chi(X)+\sum_{j=1}^n\,(\theta_j-1)<0$ and such a metric exists uniquely by the maximum principle. Both McOwen and Troyanov used PDE to show that there exists a unique conformal hyperbolic metric on $X$ representing a ${\Bbb R}$-divisor $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ with $\theta_j>0$ if and only if the above inequality holds. I would like to ask whether the condition of $\theta_j>0$ could be relaxed to $\theta_j\geq 0$ there. That is, is the following statement true or false?
$\bullet$ There exists a conformal hyperbolic metric representing a ${\Bbb R}$-divisor $D=\sum_{j=1}^n\,(\theta_j-1)\,P_j$ with $\theta_j\geq 0$ on a compact Riemann surface $X$ if $\chi(X)+\sum_{j=1}^n\,(\theta_j-1)<0$.
I believe it is correct and have been pondering over a proof for some days.
